Precise Penning trap measurements of double $\beta $-decay Q-values

The double $\beta $-decay ($\beta \beta $-decay) Q-value, defined as the mass difference between parent and daughter atoms, is an important parameter for both two-neutrino $\beta \beta $-decay (2$\nu \beta \beta$) and neutrinoless $\beta \beta $-decay (0$\nu \beta \beta$) experiments. The Q-value enters into the calculation of the phase space factors, which relate the measured $\beta \beta $-decay half-life to the nuclear matrix element and, in the case of 0$\nu \beta \beta $, the effective Majorana mass of the neutrino. In addition, the Q-value defines the total kinetic energy of the two electrons emitted in 0$\nu \beta \beta $, corresponding to the location of the single peak that is the sought after signature of 0$\nu \beta \beta $. Hence, it is essential to have a precise and accurate Q-value determination. Over the last decade, the Penning trap mass spectrometry community has made a significant effort to provide precise $\beta \beta $-decay Q-value determinations. Here we report on recent measurements with the Low Energy Beam and Ion Trap (LEBIT) facility at the National Superconducting Cyclotron Laboratory (NSCL) of the $^{48}$Ca, $^{82}$Se, and $^{96}$Zr Q-values. These measurements complete the determination of $\beta \beta $-decay Q-values for the 11 ``best'' candidates (those with Q \textgreater 2 MeV). We also report on a measurement of the $^{78}$Kr double electron capture (2EC) Q-value and discuss ongoing Penning trap measurements relating to $\beta \beta $-decay and 2EC.